810 research outputs found

    Discrete optimization methods to fit piecewise affine models to data points

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    Fitting piecewise affine models to data points is a pervasive task in many scientific disciplines. In this work, we address the k-Piecewise Affine Model Fitting with Piecewise Linear Separability problem (k-PAMF-PLS) where, given a set of m points {a1,…,am}?Rn{a1,…,am}?Rn and the corresponding observations {b1,…,bm}?R{b1,…,bm}?R, we have to partition the domain RnRn into k piecewise linearly (or affinely) separable subdomains and to determine an affine submodel (function) for each of them so as to minimize the total linear fitting error w.r.t. the observations bi.To solve k-PAMF-PLS to optimality, we propose a mixed-integer linear programming (MILP) formulation where symmetries are broken by separating shifted column inequalities. For medium-to-large scale instances, we develop a four-step heuristic involving, among others, a point reassignment step based on the identification of critical points and a domain partition step based on multicategory linear classification. Differently from traditional approaches proposed in the literature for similar fitting problems, in both our exact and heuristic methods the domain partitioning and submodel fitting aspects are taken into account simultaneously.Computational experiments on real-world and structured randomly generated instances show that, with our MILP formulation with symmetry breaking constraints, we can solve to proven optimality many small-size instances. Our four-step heuristic turns out to provide close-to-optimal solutions for small-size instances, while allowing to tackle instances of much larger size. The experiments also show that the combined impact of the main features of our heuristic is quite substantial when compared to standard variants not including them. We conclude with an application to the identification of dynamical piecewise affine systems for which we obtain promising results of comparable quality with those achieved with state-of-the-art methods from the literature on benchmark data sets

    A Framework for Generalized Benders' Decomposition and Its Application to Multilevel Optimization

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    We describe a framework for reformulating and solving optimization problems that generalizes the well-known framework originally introduced by Benders. We discuss details of the application of the procedures to several classes of optimization problems that fall under the umbrella of multilevel/multistage mixed integer linear optimization problems. The application of this abstract framework to this broad class of problems provides new insights and a broader interpretation of the core ideas, especially as they relate to duality and the value functions of optimization problems that arise in this context

    Mixed Integer Linear Programming Formulation Techniques

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    A wide range of problems can be modeled as Mixed Integer Linear Programming (MIP) problems using standard formulation techniques. However, in some cases the resulting MIP can be either too weak or too large to be effectively solved by state of the art solvers. In this survey we review advanced MIP formulation techniques that result in stronger and/or smaller formulations for a wide class of problems

    Global optimization in Hilbert space

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    We propose a complete-search algorithm for solving a class of non-convex, possibly infinite-dimensional, optimization problems to global optimality. We assume that the optimization variables are in a bounded subset of a Hilbert space, and we determine worst-case run-time bounds for the algorithm under certain regularity conditions of the cost functional and the constraint set. Because these run-time bounds are independent of the number of optimization variables and, in particular, are valid for optimization problems with infinitely many optimization variables, we prove that the algorithm converges to an (Formula presented.)-suboptimal global solution within finite run-time for any given termination tolerance (Formula presented.). Finally, we illustrate these results for a problem of calculus of variations

    Proceedings of the second "international Traveling Workshop on Interactions between Sparse models and Technology" (iTWIST'14)

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    The implicit objective of the biennial "international - Traveling Workshop on Interactions between Sparse models and Technology" (iTWIST) is to foster collaboration between international scientific teams by disseminating ideas through both specific oral/poster presentations and free discussions. For its second edition, the iTWIST workshop took place in the medieval and picturesque town of Namur in Belgium, from Wednesday August 27th till Friday August 29th, 2014. The workshop was conveniently located in "The Arsenal" building within walking distance of both hotels and town center. iTWIST'14 has gathered about 70 international participants and has featured 9 invited talks, 10 oral presentations, and 14 posters on the following themes, all related to the theory, application and generalization of the "sparsity paradigm": Sparsity-driven data sensing and processing; Union of low dimensional subspaces; Beyond linear and convex inverse problem; Matrix/manifold/graph sensing/processing; Blind inverse problems and dictionary learning; Sparsity and computational neuroscience; Information theory, geometry and randomness; Complexity/accuracy tradeoffs in numerical methods; Sparsity? What's next?; Sparse machine learning and inference.Comment: 69 pages, 24 extended abstracts, iTWIST'14 website: http://sites.google.com/site/itwist1
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